Theorem rabbiia 2850

Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)

Hypothesis

Ref	Expression
rabbiia.1	⊢ (x ∈ A → (φ ↔ ψ))

Assertion

Ref	Expression
rabbiia	⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ}

Proof of Theorem rabbiia

Step	Hyp	Ref	Expression
1		rabbiia.1	. . . 4 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.32i 618	. . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	abbii 2466	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} = {x ∣ (x ∈ A ∧ ψ)}
4		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
5		df-rab 2624	. 2 ⊢ {x ∈ A ∣ ψ} = {x ∣ (x ∈ A ∧ ψ)}
6	3, 4, 5	3eqtr4i 2383	1 ⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rab 2624

This theorem is referenced by: (None)